const In : set set prop term iIn = In infix iIn 2000 2000 term Subq = \x:set.\y:set.!z:set.z iIn x -> z iIn y term TransSet = \x:set.!y:set.y iIn x -> Subq y x term TransSet = \x:set.!y:set.y iIn x -> Subq y x term ordinal = \x:set.TransSet x & !y:set.y iIn x -> TransSet y const SNo : set prop const SNoLe : set set prop term <= = SNoLe infix <= 2020 2020 const SNoLt : set set prop term < = SNoLt infix < 2020 2020 axiom SNoLeLt_tra: !x:set.!y:set.!z:set.SNo x -> SNo y -> SNo z -> x <= y -> y < z -> x < z const SNoR : set set const add_SNo : set set set term + = add_SNo infix + 2281 2280 const SNoLev : set set const SNoS_ : set set var x:set var y:set var z:set hyp SNo x hyp SNo y hyp SNo (x + y) hyp SNo z hyp !w:set.w iIn SNoS_ (SNoLev z) -> SNoLev w iIn SNoLev (x + y) -> (x + y) < w -> (?u:set.u iIn SNoR x & (u + y) <= w) | ?u:set.u iIn SNoR y & (x + u) <= w hyp SNoLev z iIn SNoLev (x + y) hyp (x + y) < z hyp ~ ((?w:set.w iIn SNoR x & (w + y) <= z) | ?w:set.w iIn SNoR y & (x + w) <= z) claim z <= x + y -> z < z